1,424 research outputs found
Probability of local bifurcation type from a fixed point: A random matrix perspective
Results regarding probable bifurcations from fixed points are presented in
the context of general dynamical systems (real, random matrices), time-delay
dynamical systems (companion matrices), and a set of mappings known for their
properties as universal approximators (neural networks). The eigenvalue spectra
is considered both numerically and analytically using previous work of Edelman
et. al. Based upon the numerical evidence, various conjectures are presented.
The conclusion is that in many circumstances, most bifurcations from fixed
points of large dynamical systems will be due to complex eigenvalues.
Nevertheless, surprising situations are presented for which the aforementioned
conclusion is not general, e.g. real random matrices with Gaussian elements
with a large positive mean and finite variance.Comment: 21 pages, 19 figure
Stable phase retrieval with low-redundancy frames
We investigate the recovery of vectors from magnitudes of frame coefficients
when the frames have a low redundancy, meaning a small number of frame vectors
compared to the dimension of the Hilbert space. We first show that for vectors
in d dimensions, 4d-4 suitably chosen frame vectors are sufficient to uniquely
determine each signal, up to an overall unimodular constant, from the
magnitudes of its frame coefficients. Then we discuss the effect of noise and
show that 8d-4 frame vectors provide a stable recovery if part of the frame
coefficients is bounded away from zero. In this regime, perturbing the
magnitudes of the frame coefficients by noise that is sufficiently small
results in a recovery error that is at most proportional to the noise level.Comment: 12 pages AMSLaTeX, 1 figur
Real algebraic knots of low degree
In this paper we study rational real algebraic knots in . We show
that two real algebraic knots of degree are rigidly isotopic if and
only if their degrees and encomplexed writhes are equal. We also show that any
irreducible smooth knot which admits a plane projection with less than or equal
to four crossings has a rational parametrization of degree .
Furthermore an explicit construction of rational knots of a given degree with
arbitrary encomplexed writhe (subject to natural restrictions) is presented.Comment: 28 page
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